Geometric Inhibition of Superflow in Single-Layer Graphene Suggests a Staggered-Flux Superconductivity in Bilayer and Trilayer Graphene.

In great contrast to the numerous discoveries of superconductivity in layer-stacked graphene systems, the absence of superconductivity in the simplest monolayer graphene remains quite puzzling. Here, through realistic computation of the electronic structure, we identify a systematic trend that superconductivity emerges only upon alteration of the low-energy electronic lattice from the underlying honeycomb atomic structure. We then demonstrate that this inhibition can result from geometric frustration of the bond lattice that disables the quantum phase coherence of the order parameter residing on it. In comparison, upon deviation from the honeycomb lattice, relief of geometric frustration allows robust superfluidity with nontrivial spatial structures. For the specific examples of bilayer and trilayer graphene under an external electric field, such a bond-centered order parameter would develop superfluidity with staggered flux that breaks the time-reversal symmetry. Our study also suggests the possible realization of the long-sought superconductivity in single-layer graphene via the application of unidirectional strain.

Droplet tilings for rapid exploration of spatially constrained many-body systems.

Geometry in materials is a key concept which can determine material behavior in ordering, frustration, and fragmentation. More specifically, the behavior of interacting degrees of freedom subject to arbitrary geometric constraints has the potential to be used for engineering materials with exotic phase behavior. While advances in lithography have allowed for an experimental exploration of geometry on ordering that has no precedent in nature, many of these methods are low throughput or the underlying dynamics remain difficult to observe directly. Here, we introduce an experimental system that enables the study of interacting many-body dynamics by exploiting the physics of multidroplet evaporation subject to two-dimensional spatial constraints. We find that a high-energy initial state of this system settles into frustrated, metastable states with relaxation on two timescales. We understand this process using a minimal dynamical model that simulates the overdamped dynamics of motile droplets by identifying the force exerted on a given droplet as being proportional to the two-dimensional vapor gradients established by its neighbors. Finally, we demonstrate the flexibility of this platform by presenting experimental realizations of droplet-lattice systems representing different spin degrees of freedom and lattice geometries. Our platform enables a rapid and low-cost means to directly visualize dynamics associated with complex many-body systems interacting via long-range interactions. More generally, this platform opens up the rich design space between geometry and interactions for rapid exploration with minimal resources.

Alkali Transition-Metal Molybdates: A Stepwise Approach to Geometrically Frustrated Systems.

Materials with triangular arrangements of transition metal ions are of great interest for their complex magnetism resulting from geometric frustration. This paper describes the stepwise formation of kagome lattices of open shell transition-metal ions from half-delta chains to delta/sawtooth chains, and finally kagome nets. The systems can be viewed as a testbed for magnetic studies since a variety of spin states can be introduced across the same structure type, and progress through increasing levels of structural complexity and dimensionality. The synthetic and structural development of this continuum is studied here in well-formed single crystals of A2 M3 (MoO4 )3 (OH)2 (A=K, Rb; M=Mn, Co), CsM2 (MoO4 )2 (OH) (M=Mn, Fe, Co, Zn), and KM3 (MoO4 )2 O(OH) (M=Mn).

Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks.

We study two-dimensional triangular-network models, which have degenerate ground states composed of straight or randomly-zigzagging stripes and thus sub-extensive residual entropy. We show that attraction is responsible for the inversion of the stable phase by changing the entropy of fluctuations around the ground-state configurations. By using a real-space shell-expansion method, we compute the exact expression of the entropy for harmonic interactions, while for repulsive harmonic interactions we obtain the entropy arising from a limited subset of the system by numerical integration. We compare these results with a three-dimensional triangular-network model, which shows the same attraction-mediated selection mechanism of the stable phase, and conclude that this effect is general with respect to the dimensionality of the system.

Polydispersity-driven topological defects as order-restoring excitations.

The engineering of defects in crystalline matter has been extensively exploited to modify the mechanical and electrical properties of many materials. Recent experiments on manipulating extended defects in graphene, for example, show that defects direct the flow of electric charges. The fascinating possibilities offered by defects in two dimensions, known as topological defects, to control material properties provide great motivation to perform fundamental investigations to uncover their role in various systems. Previous studies mostly focus on topological defects in 2D crystals on curved surfaces. On flat geometries, topological defects can be introduced via density inhomogeneities. We investigate here topological defects due to size polydispersity on flat surfaces. Size polydispersity is usually an inevitable feature of a large variety of systems. In this work, simulations show well-organized induced topological defects around an impurity particle of a wrong size. These patterns are not found in systems of identical particles. Our work demonstrates that in polydispersed systems topological defects play the role of restoring order. The simulations show a perfect hexagonal lattice beyond a small defective region around the impurity particle. Elasticity theory has demonstrated an analogy between the elementary topological defects named disclinations to electric charges by associating a charge to a disclination, whose sign depends on the number of its nearest neighbors. Size polydispersity is shown numerically here to be an essential ingredient to understand short-range attractions between like-charge disclinations. Our study suggests that size polydispersity has a promising potential to engineer defects in various systems including nanoparticles and colloidal crystals.

Stable nematic droplets with handles.

We stabilize nematic droplets with handles against surface tension-driven instabilities, using a yield-stress material as outer fluid, and study the complex nematic textures and defect structures that result from the competition between topological constraints and the elasticity of the nematic liquid crystal. We uncover a surprisingly persistent twisted configuration of the nematic director inside the droplets when tangential anchoring is established at their boundaries, which we explain after considering the influence of saddle splay on the elastic free energy. For toroidal droplets, we find that the saddle-splay energy screens the twisting energy, resulting in a spontaneous breaking of mirror symmetry; the chiral twisted state persists for aspect ratios as large as ∼20. For droplets with additional handles, we observe in experiments and computer simulations that there are two additional -1 surface defects per handle; these are located in regions with local saddle geometry to minimize the nematic distortions and hence the corresponding elastic free energy.

Anisotropic Morphing in Bistable Kirigami through Symmetry Breaking and Geometric Frustration.

Shape morphing in bistable kirigami enables remarkable functionalities appealing to a diverse range of applications across the spectrum of length scale. At the core of their shape shifting lies the architecture of their repeating unit, where highly deformable slits and quasi-rigid rotating units often exhibit multiple symmetries that confer isotropic deployment obeying uniform scaling transformation. In this work, symmetry breaking in bistable kirigami is investigated to access geometric frustration and anisotropic morphing, enabling arbitrarily scaled deployment in planar and spatial bistable domains. With an analysis on their symmetry properties complemented by a systematic investigation integrating semi-analytical derivations, numerical simulations, and experiments on elastic kirigami sheets, this work unveils the fundamental relations between slit symmetry, geometric frustration, and anisotropic bistable deployment. Furthermore, asymmetric kirigami units are leveraged in planar and flat-to-3D demonstrations to showcase the pivotal role of shear deformation in achieving target shapes and functions so far unattainable with uniformly stretchable kirigami. The insights provided in this work unveil the role of slit symmetry breaking in controlling the anisotropic bistable deployment of soft kirigami metamaterials, enriching the range of achievable functionalities for applications spanning deployable space structures, wearable technologies, and soft machines.

Disordered magnetic ground state in a quasi-1-Dd4columnar iridate Sr3LiIrO6.

Spin-orbit coupling offers a large variety of novel and extraordinary magnetic and electronic properties in otherwise 'ordinary pool' of heavy ion oxides. Here we present a detailed study on an apparently isolated hexagonal 2Hspin-chaind4iridate Sr3LiIrO6with geometric frustration. Our structural studies reveal Li-Ir chemical order with desired stoichiometry in this compound, while x-ray absorption together with x-ray photoemission spectroscopic characterizations establish pure 5+ valence of Ir. We have established a magnetic ground state with finite Ir5+magnetic moments in this compound, contrary to the anticipated nonmagneticJeff= 0 state, through combined dc susceptibility,7Li nuclear magnetic resonance (NMR), muon spin relaxation (µSR) andab-initioelectronic structure studies. These investigations together with ac magnetic susceptibility and specific heat measurements reveal that despite having noticeable antiferromagnetic correlation among the Ir5+local moments, this system does not magnetically order down to at least 0.05 K, possibly due to geometrical exchange frustration, arising from the comparable nearest- and next-nearest-neighbor interchain Ir-O-O-Ir superexchange interaction strengths with opposite signs. However, the zero-fieldµSR analysis shows emergence of a considerable proportion of spin-freezing on top of a spin-fluctuating dynamic magnetic background down to the lowest measured temperature of 1.7 K, possibly due to some inhomogeneity and/or the much stronger intra-column Ir-Ir magnetic exchange interaction strength relative to the inter-column Ir-Ir ones. The linear temperature dependence of the magnetic specific heat (Cm) in both zero and applied magnetic fields, plus the power-law behavior of the NMR spin-lattice relaxation rate suggest a gapless spinon density of states in this charge gapped disordered magnetic ground state of Sr3LiIrO6.

Evolution of magnetic surfboards and spin glass behavior in (Fe1-pGap)2TiO5.

The unusual anisotropy of the spin glass (SG) transition in the pseudobrookite system Fe2TiO5has been interpreted as arising from an induced, van der Waals-like, interaction among magnetic clusters. Here we present susceptibility (χ) and specific heat data (C) for Fe2TiO5diluted with non-magnetic Ga, (Fe1-pGap)2TiO5, for disorder parameterp= 0, 0.11, and 0.42, and elastic neutron scattering data forp= 0.20. A uniform suppression ofTgis observed upon increasingp, along with a value ofχTgthat increases asTgdecreases, i.e.dχ(Tg)/dTg<0We also observeCT∝T2in the low temperature limit. The observed behavior places (Fe1-pGap)2TiO5in the category of a strongly geometrically frustrated SG.

Enhancement of the magnetocaloric effect in geometrically frustrated cluster spin glass systems.

In this work, we theoretically demonstrate that a strong enhancement of the magnetocaloric effect is achieved in geometrically frustrated cluster spin-glass systems just above the freezing temperature. We consider a network of clusters interacting randomly which have triangular structure composed of Ising spins interacting antiferromagnetically. The intercluster disorder problem is treated using a cluster spin glass mean-field theory, which allows exact solution of the disordered problem. The intracluster part can be solved using exact enumeration. The coupling between the inter and intracluster problem incorporates the interplay between effects coming from geometric frustration and disorder. As a result, it is shown that there is the onset of cluster spin glass phase even with very weak disorder. Remarkably, it is exactly within a range of very weak disorder and small magnetic field that is observed the strongest isothermal release of entropy.

Controllable vortex lasing arrays in a geometrically frustrated exciton-polariton lattice at room temperature.

Quantized vortices appearing in topological excitations of quantum phase transition play a pivotal role in strongly correlated physics involving the underlying confluence of superfluids, Bose-Einstein condensates and superconductors. Exciton polaritons as bosonic quasiparticles have enabled studies of non-equilibrium quantum gases and superfluidity. Exciton-polariton condensates in artificial lattices intuitively emulate energy-band structures and quantum many-body effects of condensed matter, underpinning constructing vortex lattices and controlling quantum fluidic circuits. Here, we harness exciton-polariton quantum fluids of light in a frustrated kagome lattice based on robust metal-halide perovskite microcavities, to demonstrate vortex lasing arrays and modulate their configurations at room temperature. Tomographic energy-momentum spectra unambiguously reveal massless Dirac bands and quenched kinetic-energy flat bands coexisting in kagome lattices, where polariton condensates exhibit prototypical honeycomb and kagome spatial patterns. Spatial coherence investigations illustrate two types of phase textures of polariton condensates carrying ordered quantized-vortex arrays and π-phase shifts, which could be selected when needed using lasing emission energy. Our findings offer a promising platform on which it is possible to study quantum-fluid correlations in complex polaritonic lattices and highlight feasible applications of structured light.

Triclinic Metamaterials by Tristable Origami with Reprogrammable Frustration.

Geometrical-frustration-induced anisotropy and inhomogeneity are explored to achieve unique properties of metamaterials that set them apart from conventional materials. According to Neumann's principle, to achieve anisotropic responses, the material unit cell should possess less symmetry. Based on such guidelines, a triclinic metamaterial system of minimal symmetry is presented, which originates from a Trimorph origami pattern with a simple and insightful geometry: a basic unit cell with four tilted panels and four corresponding creases. The intrinsic geometry of the Trimorph origami, with its changing tilting angles, dictates a folding motion that varies the primitive vectors of the unit cell, couples the shear and normal strains of its extrinsic bulk, and leads to an unusual Poisson effect. Such an effect, associated with reversible auxeticity in the changing triclinic frame, is observed experimentally, and predicted theoretically by elegant mathematical formulae. The nonlinearities of the folding motions allow the unit cell to display three robust stable states, connected through snapping instabilities. When the tristable unit cells are tessellated, phenomena that resemble linear and point defects emerge as a result of geometric frustration. The frustration is reprogrammable into distinct stable and inhomogeneous states by arbitrarily selecting the location of a single or multiple point defects. The Trimorph origami demonstrates the possibility of creating origami metamaterials with symmetries that are hitherto nonexistent, leading to triclinic metamaterials with tunable anisotropy for potential applications such as wave propagation control and compliant microrobots.

Frustration, strain and phase co-existence in the mixed valent hexagonal iridate Ba3NaIr2O9.

Using detailed synchrotron diffraction, magnetization, thermodynamic and transport measurements, we investigate the relationship between the mixed valence of Ir, lattice strain and the resultant structural and magnetic ground states in the geometrically frustrated triple perovskite iridate Ba3NaIr2O9. We observe a complex interplay between lattice strain and structural phase co-existence, which is typically not observed in this family of compounds. The low temperature magnetic ground state is characterized by the absence of long-range magnetic order, and points towards the condensation of a cluster glass state from an extended regime of short range magnetic correlations.

Thermodynamic Size Control in Curvature-Frustrated Tubules: Self-Limitation with Open Boundaries.

We use computational modeling to investigate the assembly thermodynamics of a particle-based model for geometrically frustrated assembly, in which the local packing geometry of subunits is incompatible with uniform, strain-free large-scale assembly. The model considers discrete triangular subunits that drive assembly toward a closed, hexagonal-ordered tubule, but have geometries that locally favor negative Gaussian curvature. We use dynamical Monte Carlo simulations and enhanced sampling methods to compute the free energy landscape and corresponding self-assembly behavior as a function of experimentally accessible parameters that control assembly driving forces and the magnitude of frustration. The results determine the parameter range where finite-temperature self-limiting assembly occurs, in which the equilibrium assembly size distribution is sharply peaked around a well-defined finite size. The simulations also identify two mechanisms by which the system can escape frustration and assemble to unlimited size, and determine the particle-scale properties of subunits that suppress unbounded growth.

Geometric frustration in the myosin superlattice of vertebrate muscle.

Geometric frustration results from an incompatibility between minimum energy arrangements and the geometry of a system, and gives rise to interesting and novel phenomena. Here, we report geometric frustration in a native biological macromolecular system---vertebrate muscle. We analyse the disorder in the myosin filament rotations in the myofibrils of vertebrate striated (skeletal and cardiac) muscle, as seen in thin-section electron micrographs, and show that the distribution of rotations corresponds to an archetypical geometrically frustrated system---the triangular Ising antiferromagnet. Spatial correlations are evident out to at least six lattice spacings. The results demonstrate that geometric frustration can drive the development of structure in complex biological systems, and may have implications for the nature of the actin--myosin interactions involved in muscle contraction. Identification of the distribution of myosin filament rotations with an Ising model allows the extensive results on the latter to be applied to this system. It shows how local interactions (between adjacent myosin filaments) can determine long-range order and, conversely, how observations of long-range order (such as patterns seen in electron micrographs) can be used to estimate the energetics of these local interactions. Furthermore, since diffraction by a disordered system is a function of the second-order statistics, the derived correlations allow more accurate diffraction calculations, which can aid in interpretation of X-ray diffraction data from muscle specimens for structural analysis.

Raman spectroscopic study of frustrated ferroelectric phase in hydroxyl salts Co2(OH)3Br/Co2(OD)3Br.

We report on an investigation of the temperature-dependent ordering of the hydrogen/deuterium atoms in geometrically frustrated magnets Co2(OH)3Br and its deuterated Co2(OD)3Br, to shed light on the origin of the newly-identified ferroelectricity using Raman spectroscopy. Significant changes in the Raman frequencies and line-widths of the Raman-active modes were observed below ∼260 K in Co2(OD)3Br and ∼240 K in Co2(OH)3Br, respectively, the analysis of which revealed strong spin-phonon couplings in this system. Further, for Co2(OD)3Br, six new phonon bands appeared below around 260 K, with the corresponding intensities obeying a power-law equationI∝1-T/Tc2ßwhereinTc= 260 K, suggesting that an ordering process occurred below ∼260 K. The ordering process subsequently affected the local structure and brought out the reported ferroelectric phase, which is considered as frustrated. Meanwhile, in Co2(OH)3Br, only one new band was observed below ∼240 K, followed by two 'softened' modes correlated to the [OH] sub-lattice below ∼185 K, wherein an incomplete ordering was suggested. The present work reveals a new multiferroic system combining geometrically frustrated magnetism and deuterium ordering-type ferroelectricity.

Quantum Disordered State of Magnetic Charges in Nanoengineered Honeycomb Lattice.

A quantum magnetic state due to magnetic charges is never observed, even though they are treated as quantum mechanical variables in theoretical calculations. Here, the occurrence of a novel quantum disordered state of magnetic charges in a nanoengineered magnetic honeycomb lattice of ultra-small connecting elements is demonstrated. The experimental research, performed using spin resolved neutron scattering, reveals a massively degenerate ground state, comprised of low integer and energetically forbidden high integer magnetic charges, that manifests cooperative paramagnetism at low temperature. The system tends to preserve the degenerate configuration even under large magnetic field application. It exemplifies the robustness of disordered correlation of magnetic charges in a 2D honeycomb lattice. The realization of quantum disordered ground state elucidates the dominance of exchange energy, which is enabled due to the nanoscopic magnetic element size in nanoengineered honeycomb. Consequently, an archetypal platform is envisaged to study quantum mechanical phenomena due to emergent magnetic charges.

Rotational switches in the two-dimensional fullerene quasicrystal.

One of the essential components of molecular electronic circuits are switching elements that are stable in two different states and can ideally be switched on and off many times. Here, distinct buckminsterfullerenes within a self-assembled monolayer, forming a two-dimensional dodecagonal quasicrystal on a Pt-terminated Pt3Ti(111) surface, are identified to form well separated molecular rotational switching elements. Employing scanning tunneling microscopy, the molecular-orbital appearance of the fullerenes in the quasicrystalline monolayer is resolved. Thus, fullerenes adsorbed on the 36 vertex configuration are identified to exhibit a distinctly increased mobility. In addition, this finding is verified by differential conductance measurements. The rotation of these mobile fullerenes can be triggered frequently by applied voltage pulses, while keeping the neighboring molecules immobile. An extensive analysis reveals that crystallographic and energetic constraints at the molecule/metal interface induce an inequality of the local potentials for the 36 and 32.4.3.4 vertex sites and this accounts for the switching ability of fullerenes on the 36 vertex sites. Consequently, a local area of the 8/3 approximant in the two-dimensional fullerene quasicrystal consists of single rotational switching fullerenes embedded in a matrix of inert molecules. Furthermore, it is deduced that optimization of the intermolecular interactions between neighboring fullerenes hinders the realization of translational periodicity in the fullerene monolayer on the Pt-terminated Pt3Ti(111) surface.

How geometric frustration shapes twisted fibres, inside and out: competing morphologies of chiral filament assembly.

Chirality frustrates and shapes the assembly of flexible filaments in rope-like, twisted bundles and fibres by introducing gradients of both filament shape (i.e. curvature) and packing throughout the structure. Previous models of chiral filament bundle formation have shown that this frustration gives rise to several distinct morphological responses, including self-limiting bundle widths, anisotropic domain (tape-like) formation and topological defects in the lateral inter-filament order. In this paper, we employ a combination of continuum elasticity theory and discrete filament bundle simulations to explore how these distinct morphological responses compete in the broader phase diagram of chiral filament assembly. We show that the most generic model of bundle formation exhibits at least four classes of equilibrium structure-finite-width, twisted bundles with isotropic and anisotropic shapes, with and without topological defects, as well as bulk phases of untwisted, columnar assembly (i.e. 'frustration escape'). These competing equilibrium morphologies are selected by only a relatively small number of parameters describing filament assembly: bundle surface energy, preferred chiral twist and stiffness of chiral filament interactions, and mechanical stiffness of filaments and their lateral interactions. Discrete filament bundle simulations test and verify continuum theory predictions for dependence of bundle structure (shape, size and packing defects of two-dimensional cross section) on these key parameters.

Synthesis and crystal structure of La21Cr8-2a Al b Ge7-b C12 [a = 0.22†(2) and b = 0.758†(19)].

Single crystals of a new multinary chromium carbide, La21Cr8-2a Al b Ge7-b C12 (henicosa-lanthanum octa-chromium aluminium hexa-germanium dodeca-carbide), were grown from an La-rich self flux and were characterized by single-crystal X-ray diffraction. The face-centered cubic crystal structure is composed of isolated and geometrically frustrated regular Cr tetra-hedra that are co-centered within regular C octa-hedra. These mutually separated Cr4-aC6 clusters are distributed throughout a three-dimensional framework of Al, Ge, and La. The title compound is isotypic with La21-δMn8X7C12 and R21Fe8X7C12 (R = La, Ce, Pr; X = Al, Bi, Ge, Sn, Sb, Te) and represents the first example of a Cr-based compound with this structure-type.